Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.

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    $\begingroup$ Let p(E) denote the total (integral) Pontryagin class of a real bundle E. It's not too hard to show that p(E (+) F) = p(E) * p(F) mod 2-torsion. The 2-torsion term is stated in Theorem 1.6 of Brown's "The Cohomology of BSO_n and BO_n with Integer Coefficients". $\endgroup$ – skd May 22 at 4:23
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    $\begingroup$ @sdk: Brown's paper is about vector bundles, while my question is about locally trivial bundles with fiber homeomorphic to $\mathbb R^n$. $\endgroup$ – Igor Belegradek May 22 at 12:40
  • $\begingroup$ Ah, yes. Sorry, didn't read carefully enough. $\endgroup$ – skd May 22 at 17:29

Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.

Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (stable, topological) bundles. The question is whether the identity $$\mu^* p_n = \sum_{ a + b = n} p_a \otimes p_b$$ holds. As the map $BO \to BTOP$ is a rational equivalence and an H-space map, it is equivalent to verify this equation in the cohomology of $BO$ instead, where it indeed holds.

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    $\begingroup$ Thank you! For my record here is a reference: on p.215 of Boardman-Vogt's book "Homotopy invariant algebraic structures on topological spaces" the authors say (roughtly) that since $O\to Top\to F$ are topological monoid homomorphisms under the Whitney sum, these maps are infinite loop space maps, and hence so are the corresponding maps of classifying spaces $BO\to BTop\to BF$. Actually, instead of this last sequence they write $BO\to Top\to F$ which I assume is a misprint. $\endgroup$ – Igor Belegradek May 22 at 12:29
  • $\begingroup$ Is it true more generally that for any topological monoid (or at least group) homomorphism $G\to H$ the corresponding map of classifying spaces $BG\to BH$ is an infinite loop map? $\endgroup$ – Igor Belegradek May 22 at 12:29
  • $\begingroup$ @ConnorMalin: thank you, I see now that Boardman-Vogt's result has quite a few technical assumptions; they speak of symmetrical monoidal functors. I gather, the point is that Whitney sum is commutative. $\endgroup$ – Igor Belegradek May 22 at 13:05
  • $\begingroup$ Sorry, I deleted my comment in fear that I was missing something. For posterity, I said that usually a group will not be an infinite loop space. $\endgroup$ – Connor Malin May 22 at 13:35
  • $\begingroup$ @Oscar: actually, I am confused. It seems your argument only proves that topological Pontryagin classes satisfy the Whitney sum formula on vector bundles. I need it for topological bundles. Basically, there are two H-maps which are rational equivalences from $BSO$: one to $BSTOP$ and the other one to the product of $K(\mathbb Z, 4i)$ spaces where $i$ varies over positive integers. The latter map is the integral Pontryagin class. One needs to show that the topological Pontryagin class completes the square rationally. None of the maps from $BSO$ can be reversed, I think. $\endgroup$ – Igor Belegradek May 22 at 22:49

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